From what I can tell from their website, ffmpeg is basically a file converter and player of mutlimedia files. Can you sync up the audio to your video? Can you do any video or audio editing? If not, what do you use for those duties, if anything?

Ffmpeg is a lot more than that. It can basically take just about any audio/video input source be it hardware or an existing file and mux/demux it. In my case I instruct ffmpeg to take my audio interface as audio track source, my webcam as video track source, set both to lossless formats and output into MPG file format as an example. It's an advanced piece of software and may need a little effort to get used to, but it's worth it.

Youtube, VLC Player, Chrome, Firefox are using ffmpeg just to name a few.

If you are using the camera audio, what is the device on the other tripod?

As I posted "sometimes using just the camera audio..."
Depending on what I'm doing I do use the H1 or a similar small recorder and combine them in a editing program. I currently use a Tascam DR-05 that I much prefer.

For combining video and audio I've used Windows Moviemaker in the past, but my latest version of Windows no longer had Moviemaker built in.

I worked with ffmpeg for a while but it has a HIGH learning curve. It's geared much more towards someone who is doing professional editing.

Reaper will handle simple projects easily. Fran Guidry has a nice tutorial for doing exactly that on his Homebrewedmusic.com website.

I would like to get more serious about the video thing, but I really don't have a reason to do it at present. I find the fisheye effect really anoying in all the standard fare for shooting simple videos and the audio is pretty nasty when the mic's back a few feet, so I might as well stick with the point and shoot until I find a reason to move to something else.

If you have the money I would suggest buying a DSLR camera.
The panasonic g7 seems like an amazing deal. 1080p up to 60fps, and 4k 30 fps, with a flippable screen, and several other uses
Don't know about the tascam, but the zoom h5 has high quality, and has a line out that you can plug in to provide the audio for the camera, it also has 2 inputs with 48v phantom power if you want to upgrade further.
After recording you can just preview the video on the camera with headphones.
If you like it tranfer to pc, add the effect you want with either lightworks or shotcut, and profit.

I currently have a pretty simple setup, using my old ZOOM Q3HD for both audio and video. I separate the audio with Audacity (free) to get rid of background noise, use Mp3 Gain (free) to set the volume to the level I use when creating mp3s from my old vinyl, then reassemble the audio and video and clip both in HitFilm Express (also free and recommended by Fran Guidry). I still have MovieMaker on my PC so use that to create the final video.

The biggest downside is the inability to get a good balance between vocals and guitar, and while I've thought about getting a new audio interface to record separate tracks (my old one doesn't work with newer operating systems), I'm not sure the result would be worth the added effort and expense.